# A loop quantum gravity toy inspired by an Aharonov-Bohm ring

Comparing my question to Give a description of Loop Quantum Gravity your grandmother could understand what I'm looking for here is a toy for a toddler ($\approx$ a pre-QFT graduate student).

I seek to generalize to gravity the following minimalistic model of magnetism (which is a textbook example for the phenomena of Aharonov-Bohm oscillations and persistent currents).

Take a three-state Hilbert space with a Hamiltonian matrix $h_{nm}$ with $n,m=1,2,3$. Interpreting these states as localization sites for, say, an electron in a mesoscopic ring, one may ask how to include a magnetic field piercing the ring. The answer is given by the gauge-invariant phase $\varphi =\arg (h_{12} h_{23} h_{13}^{\ast})$ which is the Aharonov-Bohm phase for a closed trajectory (1-2-3) along the ring; $\varphi$ is equal to the magnetic flux through the ring expressed in units of $\phi_0=h/e$. If $\varphi \not = 0 (\rm{mod} 2 \pi)$, then the lowest energy state carries persistent current, and the model serves as a good starting point to discuss this piece of mesoscopic physics.

I thinks it would be great to have such a toy example for gravity (classical and quantum). The can probably found in be found in a pedagogical book, so please give a reference if you know one.

More specifically, my question would be:

What is the minimal number of points to define a meaningful finite-difference analogue of Hilbert-Einstein action and how to do that? Can basic ideas of LQG quantization be demonstrated on such a finite state model?

I am looking for a simple, possibly less than $3+1$-dimensional example to demonstrate the possibility to eliminate gravity by local but not global Lorentz transformations the same way the above toy model demonstrates the relation between local and global $U(1)$ symmetry.